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Derivation and Regularization of Corrections to Propagators and Vertices for Cubically Interacting Scalar Field Theories

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Derivation and Regularization of Corrections to Propagators and Vertices for Cubically Interacting Scalar Field Theories DERIVATION AND REGULARIZATION OF CORRECTIONS TO PROPAGATORS AND VERTICES FOR CUBICALLY INTERACTING SCALAR FIELD THEORIES A Thesis by Richard Traverzo Bachelor of Science, Wichita State University, 2010 Submitted to the Department of Mathematics, Statistics, and Physics and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Master of Science May 2013 c Copyright 2013 by Richard Traverzo All Rights Reserved DERIVATION AND REGULARIZATION OF CORRECTIONS TO PROPAGATORS AND VERTICES FOR CUBICALLY INTERACTING SCALAR FIELD THEORIES The following faculty members have examined the final copy of this thesis for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Master of Science with a major in Mathematics. Thomas DeLillo, Committee Chair Holger Meyer, Committee Member Thalia Jeffres, Committee Member iii ABSTRACT This thesis, a mainly expository work, attempts to give all the calculational details for some popular models in scalar quantum field theory that serve a mostly pedagogical purpose. Starting with a real scalar field with a given Lagrangian in classical field theory, standard quantization results given in most introductory QFT texts are derived with as many explicit calculations (which are often left to the reader) as possible. After presenting this development, we gather specific calculations that model scattering and decay processes. In many texts, the author(s) simply gives one or two calculations as an example. Here we organize several different calculations in a single work. iv ACKNOWLEDGEMENTS There are many people who I would like to thank for helping make this thesis possible. First let me extend my gratitude to Dr. Tom DeLillo, my thesis advisor, for his direction and insights over the course of the past couple of years. The other members of my thesis committee, Dr. Thalia Jeffres and Dr. Holger Meyer, also deserve recognition for their participation in the process of my oral defense. Other faculty members who offered friendly and useful advice were Dr. Abdulhamid Albaid, Dr. Elizabeth Behrman, and Dr. Jason Ferguson, and I am grateful for having recieved such advice. For personal support I extend my thanks to my family: my mother Wanda, uncle Leo, aunt Betty, grandmother Angelita, and grandfather Leonardo. Thank you to my girlfriend Ariana for her continued support throughout this process. Of course the greatest thanks goes to God who makes all things possible. v PREFACE The idea for this masters thesis originated out of Tom DeLillo's 2009 Quantum Field Theory course. He suggested that a good possible topic for the preliminary calculations and formalism for scalar quantum field theory. Complete details for most theories are often scattered over several sources, therefore one objective of this thesis is to lay out all the calcu- lational details of a popular model, or "toy" theory, for students of physics or mathematics as preparation for more complicated physical theories (like Quantum Electrodynamics) in- cluding spin or charge. These details are also laid out as reference for a more rigorous mathematical development of QFT; see eg. [1][2]. These calculations when combined into a manuscript could also be used as a guide for a student embarking on a course in QFT or even be used as a set of notes accompanying the standard texts of the field. With this in mind, the reader is assumed to be familiar with introductory quantum mechanics at the level of [3]. Also some basic special relativity is required. Though often presented in a tensor notation we forego this method in a general sense. Four-vectors, how- µ ever, will be denoted in the typical sense: xµ = (x0; −x1; −x2; −x3) and x = (x0; x1; x2; x3) which are given by the metric signature (1; −1; −1; −1). In the introductory development µ 2 2 2 2 we make use of the Einstein summation convention x xµ = x0 − x1 − x2 − x3. vi TABLE OF CONTENTS Chapter Page 1 INTRODUCTION...................................1 2 CLASSICAL FIELD THEORY............................3 2.1 Lagrangian Mechanics; the Euler Lagrange Equations.............3 2.2 The Euler-Lagrange Equations for Fields....................5 2.3 Hamiltonian Mechanics..............................6 2.4 The Real Scalar Field...............................7 2.5 A Scalar Field Theory with Interactions....................8 3 QUANTIZATION OF THE CLASSICAL KLEIN-GORDON FIELD....... 10 3.1 Obtaining the Hamiltonian and Momentum.................. 13 3.2 Constructing the Fock Space and Evaluating the Energy and Momentum of its Elements.................................... 15 3.3 Obtaining a Time-Dependant Form of φ .................... 20 3.4 Quantizing the ABB and ABC-Theory..................... 22 3.5 Calculating h0jap; φ(x)j0i ............................. 23 3.6 Derivation of the Feynman Propagator..................... 25 4 PERTURBATION THEORY............................. 29 4.1 Deriving the perturbation expansion...................... 29 4.2 Expressing φ(x) in Terms of φI (x)........................ 30 5 TREE-LEVEL AMPLITUDES IN ABC AND ABB THEORY........... 32 5.1 AB-Scattering in ABC Theory.......................... 32 5.1.1 The Zeroth Order Term in Expansion of g ............... 32 5.1.2 The First Order Term in Expansion of g ................ 33 5.1.3 The Second-Order Term in Expansion of g ............... 35 5.2 C ! A + B Decay................................ 43 5.3 A ! B + B Decay................................ 47 6 DIMENSIONAL REGULARIZATION........................ 52 6.1 One-loop Correction to the A-Propagator in ABB Theory........... 52 6.2 One-loop Correction to the Vertex in ABC Theory............... 55 6.3 Fourth-Order Self Energy Diagram: Nested Divergence............ 56 6.4 Fourth-Order Self-Energy Diagram: Overlapping Divergence......... 59 6.4.1 Calculating the Boundary Contribution of the dη Integral....... 62 6.5 Calculating the integral contribution to the dη Integral............ 64 vii TABLE OF CONTENTS (continued) Chapter Page 7 CONCLUSION AND FUTURE WORK....................... 68 REFERENCES....................................... 71 viii LIST OF FIGURES Figure Page 5.1 Zeroth-order term in the power series expansion of g............... 33 0 5.2 C-Exchange in AB-scattering. The qA refers to the pA in the discussion and similarly for qB.................................... 37 0 5.3 AB Annihilation for ABC Theory. qA corresponds to pA and similarly for qB.. 41 5.4 A C-particle decaying into an A and a B-particle................. 43 5.5 An A-particle decaying into 2 B-particles..................... 47 6.1 Self-Energy of The A-Particle in ABB Theory................... 53 6.2 One-loop correction to the vertex of ABC theory................. 55 6.3 Two-loop correction to the A-propagator: nested divergence........... 57 6.4 Two-loop correction to the propagator: overlapping divergence......... 59 ix CHAPTER 1 INTRODUCTION Quantum field theory is the attempt to combine quantum mechanics with special relativity. In quantum mechanics, the (non-relativistic) Schrodinger equation which is second order in space and first order in time gives rise to solutions which are interpreted as having positive energy. Unfortunately when we try to put forward this same interpretation in the relativistic case (an equation both second order in space and time) we obtain \negative energy" solutions. To ameliorate this issue we extend relativistic quantum mechanics to relativistic quan- tum field theory. φ is then interpreted not as a wavefunctions for a certain particle but as a field for infinitely many particles. Chapter 2 introduces field theory in the classical sense in preparation for quantization of the fields which takes place in Chapter 3. For this introduc- tory material I follow mainly Peskin and Schroeder [4], although some material (especially the derivation of the Euler-Lagrange equation) is contained in Maggiore [5]. After the introductory development and a brief explanation of perturbation theory (in Chapter 4), I present some basic interacting theories which are called the ABC and ABB-theories. The Feynman rules for ABC are stated in [6] and [7] and derived in [8]. The Feynman rules for the ABB theory are stated in [9], and to our knowledge have not been derived from a Lagrangian. We state this Lagrangian and derive the Feynman rule for a vertex. Chapter 5 is devoted to calculating scattering amplitudes, or S-matrix elements. In the laboratory any n number of particles collide with one another and turn into m (possibly different) particles. The entries of the S-matrix hnjSjmi can be written in a perturbative expansion of some interaction parameter g and may be interpreted as the probability am- plitude for the initial state to evolve into the particular final state. These probabilities are 1 physically measured by decay rates Γ or scattering cross-sections σ both of which will not be considered here. This process of calculating amplitudes through evaluation of S-matrix elements works fine until higher-order contributions to the expansion are calculated. It turns out that almost all higher order calculations yield infinite results. We must then reinterpret what we mean by the parameters of the theory (mass, coupling, and field.) This is the subject of Chapter 6. Through a process known as dimensional regularization we are able to isolate the singular part of the integral and recover the finite part, discarding infinitesimal contributions. Covariant regularization is another method for isolating these infinite parts of the integrals (see [10]). It yields the same finite results (with different singular parts), but it will not be considered here. 2 CHAPTER 2 CLASSICAL FIELD THEORY Before we are able to introduce quantum field theory, we need to first review the basics of classical mechanics. [11] formulates mechanics in a number of useful ways including Poisson brackets and Hamilton-Jacobi theory (both of which are useful in the transition from classical to quantum mechanics.) Of most importance here, however, will be Lagrangian and Hamiltonian mechanics. We will first introduce the Lagrangian then define the Hamiltonian from it. In future chapters we will rely mainly on the Hamiltonian formalism.
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